186 XIII INTEGRATION End of the Proof (13.14.1.7) For every function veS such that/^ v, we have^ g vg, and f* v dv = I* vg dp by (13.14.1.2), so that $* fg dp £ J* vg d^ = J* v dv. Hence, by definition of the upper integral, we have J* fg d\i g J*/^v, Hence it remains to establish the opposite inequality (13.14.1.8) |/dvg \*fgdfj.. Let h e«/ be such that h^fg. Then it is enough to show that (13.14.1.9) \*fdv£ hdiJL. The set X — A is the union of a denumerable increasing sequence of compact sets Hn and a ju-negligible set N, such that g \ Hn is continuous, finite and >0 for all n. We define a mapping u of X into R as follows: u = h/g in the union of the sets Hn, and u(x) = + oo in N and in A. In each of the sets Hn we have ug = h, and by virtue of (13.14.1.6) r* r** u dv = w JHn jHn But v(N) = 0 by virtue of (13.14.1.3), and v(A) = 0 by virtue of (13.14.1.5), hence, as h ^ 0, (13.5.7) r* p* r* r* u dv = sup u dv = sup /i dju g /i djU. J n JHM n jHn J Since/^w, we obtain the required inequality (13.14.1.9). Q.E.D. (13.14.2) Let g be a locally ii-integrable function 'which is ^0 on X, /ef S be the ^-measurable set of points xeX such that g(x) > 0, and let v = g • /i. Le^ /6e a mapping ofX into R. Then the following conditions (with the conventions of (13.11) for products) are equivalent: (a) / is v-measurable ; (b) /<ps w \n-measurable; (c) y<7 z,y \i-measurable. We may suppose # to be finite. With the conventions we have made, we have jfc = (f(psK09s)> and since the two factors on the right-hand side neverp, = \ fg d^